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Question: Find the cube of the following binomial expressions: \(\left( {4 - \dfrac{1}{{3x}}} \right)\)...

Find the cube of the following binomial expressions:
(4−13x)\left( {4 - \dfrac{1}{{3x}}} \right)

Explanation

Solution

Hint – In this question we simply need to find the cube of the given binomial expression so simply use the direct formula for (a−b)3=a3−b3−3a2b+3ab2{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2} to get the answer.

Complete step-by-step answer:
Given binomial expression is
(4−13x)\left( {4 - \dfrac{1}{{3x}}} \right)
Now we have to find out the cube of this expression.
⇒(4−13x)3\Rightarrow {\left( {4 - \dfrac{1}{{3x}}} \right)^3}
Now as we know (a−b)3=a3−b3−3a2b+3ab2{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2} so, use this property in above equation and expand we have,
⇒(4−13x)3=43−(13x)3−3(4)2(13x)+3(4)(13x)2\Rightarrow {\left( {4 - \dfrac{1}{{3x}}} \right)^3} = {4^3} - {\left( {\dfrac{1}{{3x}}} \right)^3} - 3{\left( 4 \right)^2}\left( {\dfrac{1}{{3x}}} \right) + 3\left( 4 \right){\left( {\dfrac{1}{{3x}}} \right)^2}
Now simplify the above equation we have,
⇒(4−13x)3=64−127x3−16x+43x2\Rightarrow {\left( {4 - \dfrac{1}{{3x}}} \right)^3} = 64 - \dfrac{1}{{27{x^3}}} - \dfrac{{16}}{x} + \dfrac{4}{{3{x^2}}}
So, this is the required cube of the given binomial expression.

Note – Whenever we face such types of problems the key concept is to have the basic understanding of the direct algebraic formula for (a−b)3{\left( {a - b} \right)^3}. The gist of direct algebraic formula helps in direct simplification of the given problem statement.